More about Solving D-9 | HP 33s Scientific specs

f (x)

Case Where No Root Is Found

Example: A Relative Minimum.

Calculate the root of this parabolic equation:

x2 – 6x + 13 = 0.

It has a minimum at x = 3.

Enter the equation as an expression:

Keys:	Display:	Description:
H		Selects Equation mode.
LX 2		Enters the equation.
6 zLX
13	%:. º%-
	/	Checksum and length.
	/

More about Solving D–9

Page 333

Image 333

HP 33s Scientific manual More about Solving D-9, Example a Relative Minimum

Contents

Hp 33s scientific calculator Printing History Contents Clearing the X-Register RPN The Automatic Memory Stack Storing Data into Variables How You Can Use Equations Summary of Equation Operations Fractions in Programs Solving an Equation Complex Stack Complex Operations ContentsFor More Information Accuracy of Integration Programming Data Input and Output 12-4 Entering a Program 12-5 12-412-6 12-7 13-2 13-113-3 Branching GTO 13-4 Part 3. Appendixes and Reference Managing Calculator Memory ContentsVector Operations 15-1 15-12 Round-Off Error Contents Resetting the Calculator Clearing MemoryDisabling Operations Neutral Operations About ALG Underflow How the Integral Is EvaluatedConditions That Prolong Calculation Time Page Basic Operation PartPage Getting Started Adjusting Display ContrastImportant Preliminaries Turning the Calculator On and Off Highlights of the Keyboard and Display Shifted Keys2Getting Started Cursor Keys Alpha Keys Silver Paint Keys Backspacing and Clearing4Getting Started Are described in Using Menus on page 1-7. If Keyboard-entry modeEquation-entry mode ¾ when number entry is complete 6Getting Started HP 33s Menus Chapter Name Using MenusProgramming Instructions Numeric Functions 8Getting Started Other functionsDescription To use a menu function KeysDisplay Exiting Menus To select RPN mode RPN mode ALG modeTo select ALG mode RPN and ALG Keys First Line Second Line Display and Annunciators 12Getting Started HP 33s Annunciators Meaning Chapter2 3 RAD or §,¨ Making Numbers Negative Keying in NumbersExponents of Ten Keys Display Description Understanding Digit Entry Keying in Exponents of TenOther Exponent Functions Range of Numbers and Overflow Doing ArithmeticKeys Display 16Getting Started Two-Number Functions One-Number Functions, \, , Qor T Key in the number. You dont need to press ‘ Periods and Commas in Numbers Controlling the Display Format‘3 z D, , \, ,Q Number of Decimal Places Fixed-Decimal Format %Scientific Format 20Getting Started SHOWing Full 12-Digit PrecisionEngineering Format ALL Format Fractions Entering FractionsKeysDisplayDescription 22Getting Started Messages Displaying FractionsDisplayDescription Checking Available Memory Calculator MemoryClearing All of Memory 24Getting Started RPN The Automatic Memory Stack What the Stack IsRPN The Automatic Memory Stack X and Y-Registers are in the Display Clearing the X-Register2RPN The Automatic Memory Stack Reviewing the Stack ¶ Roll DownRµ Roll Up Arithmetic How the Stack Does It Exchanging the X- and Y-Registers in the Stack4RPN The Automatic Memory Stack Lost How Enter Works How Clear x Works Using a Number Twice in a Row6RPN The Automatic Memory Stack Filling the stack with a constant Last X Register , \, , Qor T Correcting Mistakes with Last8RPN The Automatic Memory Stack Wrong One-Number Function › Reusing Numbers with LastMistake 96.704 + 10RPN The Automatic Memory Stack Work from the Parentheses Out Chain Calculations in RPN mode 12RPN The Automatic Memory Stack Exercises SolutionOrder of Calculation 16.3805x5 = 181.0000 More Exercises 14RPN The Automatic Memory Stack‘14 ‘ ⋅ 0.8 ⎟ 12.5 − 0.73 = Page Storing Data into Variables Storing Data into Variables 2Storing Data into Variables Storing and Recalling NumbersTo recall a copy of a number from a variable to the display Example Storing Numbers Viewing a Variable without Recalling It Reviewing Variables in the VAR CatalogTo review the values at any or all non-zero variables Arithmetic with Stored Variables Clearing VariablesStorage Arithmetic 4Storing Data into Variables Result 3 That is, x Recall Arithmetic Exchanging x with Any Variable LzE6Storing Data into Variables Variable Page Real-Number Functions Exponential and Logarithmic FunctionsReal-Number Functions Quotient and Remainder of Division Power FunctionsTo Calculate Press 2Real-Number Functions Trigonometry Entering πTo Calculate Press Result Trigonometric Functions Setting the Angular ModeOption Description Annunciator 4Real-Number Functions Programming Note NzR Percentage Functions Hyperbolic Functions6Real-Number Functions ‘x Q 16.12 ‘ 15.76 T . Physics Constants Const Menu Items Description Value8Real-Number Functions Conversion Functions Items Description ValueTo insert a constant Coordinate Conversions 10Real-Number FunctionsTo convert between rectangular and polar coordinates Example Conversion with Vectors Example Polar to Rectangular Conversion 12Real-Number Functions Time ConversionsTo convert between decimal fractions and minutes-seconds Example Converting Time Formats Unit Conversions Angle ConversionsTo Convert Press Displayed Results To convert an angle between degrees and radians Factorial Probability FunctionsGamma Probability Keys Example Combinations of PeopleDisplay 16Real-Number Functions Parts of Numbers Names of Functions Page Entering Fractions Fractions Fractions in the Display Display Rules2Fractions Entered Value Internal Value Displayed Fraction Accuracy Indicators Changing the Fraction Display Longer Fractions4Fractions Choosing a Fraction Format Setting the Maximum Denominator To Get This Fraction Format Change These Flags Examples of Fraction DisplaysFraction How 2.77 Is Displayed Format 6Fractions Rounding Fractions Fraction Number Entered and Fraction Displayed Format ¼Denominator Fixed 8Fractions Fractions in Equations Fractions in Programs Page How You Can Use Equations Entering and Evaluating EquationsEntering and Evaluating Equations Example Calculating with an Equation ZNz ZLL2Entering and Evaluating Equations Enters and leaves Equation mode Summary of Equation OperationsKey Operation Variables in Equations Entering Equations into the Equation List4Entering and Evaluating Equations To enter an equation Functions in Equations Numbers in Equations Parentheses in Equations Displaying and Selecting EquationsLR d ZLC z ——— Editing and Clearing EquationsTo view a long equation To select an equation 8Entering and Evaluating Equations Evaluating Equations Types of Equations 10Entering and Evaluating Equations Type of Equation Result for ‘Fx ¼ To evaluate an equation Example Evaluating an Equation with Enter Using Enter for Evaluation Responding to Equation Prompts Using XEQ for Evaluation12Entering and Evaluating Equations Example Evaluating an Equation with XEQ Operator Precedence Syntax of Equations‘5 g „ To display digits hidden by the prompt, press Order Operation Example Equations Meaning14Entering and Evaluating Equations Σxy Equation Functions 16Entering and Evaluating Equations Operation Equation functionExample Perimeter of a Trapezoid Perimeter = a + b + h Cos π / n HLA ’.25 zLN zLD 2 zR NqLN `qONqLN `‘Example Area of a Polygon Area = Verifying Equations Syntax Errors18Entering and Evaluating Equations Example Checksum and Length of an Equation Solving an Equation Solving EquationsSolving Equations Markup ⋅ Cost = Price ZLT › LD dLV2Solving Equations Example Solving the Equation of Linear Motion Value ZL T HL P z4Solving Equations Example Solving the Ideal Gas Law Equation Understanding and Controlling Solve LV q 6Solving Equations Verifying the Result Choosing Initial Guesses for Solve Interrupting a Solve Calculation 8Solving Equations Example Using Guesses to Find a Root 40 H ZL H 10Solving Equations For More Information Page Integrating Equations Integrating Equations Integrating Equations ≥ FN J0 x = π1 ≥0π cos x sin t dt2Integrating Equations Value ≥ 0π f t 4Integrating Equations Example Sine Integral= ≥ t Accuracy of Integration Interpreting Accuracy Specifying Accuracy6Integrating Equations Example Specifying Accuracy Example Changing the Accuracy Its uncertainty indicates 8Integrating Equations Complex Stack Operations with Complex Numbers+ iy To enter a complex number Complex Operations 2Operations with Complex NumbersTo do an operation with one complex number To do an arithmetic operation with two complex numbers Functions for One Complex Number, z To Calculate Press ‘2 ‘ ‘4 2 5 ‘4Operations with Complex Numbers 2 3 ‘ Using Complex Numbers in Polar Notation‘2 Example Vector Addition 6Operations with Complex Numbers Operations with Complex Numbers Page Base Menu Base Conversions and ArithmeticMenu label Description Base Conversions and Arithmetic 10-2Base Conversions and Arithmetic Arithmetic in Bases 2, 8, Keys Display Negative Numbers Representation of Numbers˜˜ 10-4Base Conversions and Arithmetic 7FFFFFFFF Range of Numbers 10-6Base Conversions and Arithmetic Windows for Long Binary Numbers Statistical Operations Entering Statistical DataStatistical Operations Entering One-Variable Data Correcting Errors in Data EntryEntering Two-Variable Data 11-2Statistical Operations Initial x, y Corrected x, y 400‘20 Mean Statistical CalculationsStatistics Menus Key Description 11-4Statistical Operations · Example Weighted Mean Two VariablesPrice per Part Number of Parts y Population Standard Deviation Sample Standard DeviationUº 11-6Statistical Operations σº Linear RegressionKeys DisplayDescription Linear Regression Menu Example Curve Fitting 11-8Statistical OperationsNitrogen Applied Grain Yield 70, y Limitations on Precision of Data Summation Statistics Summation Values and the Statistics Registers11-10Statistical Operations Normalizing Close, Large Numbers Statistics Registers in Calculator Memory Access to the Statistics RegistersExample Viewing the Statistics Registers # ™ ´º¸/ ™ ´¸/ ™ ´º/ ™ ´¸/ ™ ´º/ ‡ 11-12Statistical Operations Programming Page Simple Programming RPN mode 5 !Nz ALG mode 5 !zN‘Simple Programming Example a Simple Program RPN mode RPN mode ALG modeDisplay Description 12-2Simple Programming Designing a Program Selecting a ModeProgram Boundaries LBL and RTN Program Labels Data Input and Output Using RPN, ALG and Equations in ProgramsStrengths of RPN Operations ALG Operations To enter a program into memory Entering a Program Keys That Clear 12-6Simple ProgrammingTo enter an equation in a program line RPN mode Function Names in ProgramsExample Entering a Labeled Program ZLR HN 12-8Simple Programming Example Entering a Program with an Equation Running a Program Testing a ProgramExecuting a Program XEQ NzX 12-10Simple Programming Example Testing a Program Using Input for Entering Data Entering and Displaying Data@ To use the Input function in a program 12-12Simple Programming To respond to a prompt Using View for Displaying Data Using Equations to Display Messages 12-14Simple ProgrammingExample INPUT, VIEW, and Messages in a Program LR z › LH NzLR ›p = 2π R2 + 2π RH = 2π R R + H 12-16Simple Programming Displaying Information without Stopping Stopping or Interrupting a Program Error StopsProgramming a Stop or Pause STOP, PSE Interrupting a Running Program 12-18Simple Programming Editing a ProgramTo delete a program line To insert a program line Viewing Program Memory Program Memory Catalog of Programs MEM Memory UsageClearing One or More Programs 12-20Simple Programming To clear all programs from memory Checksum Nonprogrammable Functions Selecting a Base Mode in a ProgramProgramming with Base V Polynomial Expressions and Horners Method Decimal mode set Hexadecimal mode setNumbers Entered in Program Lines Ax4 + Bx3 + Cx2 + Dx + E Keys Display ALG mode ‰X12-24Simple Programming Checksum and length E41A Page Programming Techniques Routines in ProgramsProgramming Techniques 13-2Programming Techniques Calling Subroutines XEQ, RTN Example a Nested Subroutine Nested Subroutines Branching GTO 13-4Programming TechniquesA2 + B2 Programmed GTO Instruction Using GTO from the Keyboard 13-6Programming Techniques Conditional Instructions Test Menus Tests of Comparison x?y, x?0 Flags Program Lines Description RPN mode13-8Programming Techniques !- % ! ! ! º¸@ ! Flag Fraction-Control Flags Status ClearSet 13-10Programming Techniques Using Flags Flags Menu Menu Key DescriptionAnnunciators for Set Flags 13-12Programming Techniques Example Using Flags Program Lines Description RPN mode $ @ $ $ @ $ Example Controlling the Fraction Display 13-14Programming Techniques Program Lines Description 13-16Programming Techniques Loops Conditional Loops GTO ! ! . ! º6¸@ ! #$ ! Loops with Counters DSE, ISG 13-18Programming TechniquesLoop-Control Number $ $ ! $$ Indirectly Addressing Variables and Labels Variable13-20Programming Techniques If i contains Then i will address Indirect Address 13-22Programming Techniques Program Control withExample Choosing Subroutines With ! L Program lines If i holds Then XEQi calls- L & %1L2 Example Loop Control With 13-24Programming Techniques Equations with L 1L2 Page Solving a Program Solving and Integrating ProgramsSolving and Integrating Programs To solve a programmed function Example Program Using ALG 14-2Solving and Integrating ProgramsV= N x R x T Pressure ⋅ volume. Pressure ⋅ volume Example Program Using Equation LP z LV ’ LN z LR z LT ‘14-4Solving and Integrating Programs @ @ ‘10 !@ # / 14-6Solving and Integrating Programs Using Solve in a Program/ label #variable Example Solve in a Program Program Lines In RPN mode Integrating a ProgramTo integrate a programmed function To write a program for ≥ FN 14-8Solving and Integrating Programs Using Integration in a Program Sin/ label 14-10Solving and Integrating Programs ≥ G variableExample ≥ FN in a Program Restrictions on Solving and Integrating Page Vector Operations Mathematics ProgramsMathematics Programs = R cosP = arctan 15-2Mathematics Programs = XU + YV + ZWV1 = X i + Y j + Z k V2=U i + V j + W k Program Listing X2 + Y2 + Z2 and P 15-4Mathematics Programs Program Lines Description ALG mode Goes to the vector-addition routine 15-6Mathematics Programs Flags Used RemarksProgram Instructions Variables Used 15-8Mathematics Programs XR @ 15-10Mathematics Programs 07 g !@ 125 g @ Solutions of Simultaneous Equations 15-12Mathematics ProgramsAX + DY + GZ = J BX + EY + HZ = K CX + FY + IZ = L % 15-14Mathematics Programs Mathematics Programs 15-16Mathematics Programs Flags Used Scratch 15-18Mathematics Programs @ 15-20Mathematics Programs Polynomial Root Finder@ @ @ X2 + J + Lx + K + M = Where J = a3/2 = y0 /2X2 + J Lx + K M = ⋅ the sign of JK a1/2 15-22Mathematics Programs Polynomial ! 15-24Mathematics Programs B0 =a04a0 a32 a12 Checksum and length C7A6B2= -a2 B1 = a3a1 4a0 JK. a1 15-26Mathematics Programs + L + M 15-28Mathematics Programs Order Terms and Coefficients 15-30Mathematics Programs %/ 15-32Mathematics Programs Coordinate Transformations Old coordinate System 15-34Mathematics Programs Checksum and length 8C82 Remark 15-36Mathematics Programs Ÿ @ 15-38Mathematics Programs Statistics Programs Curve FittingStatistics Programs 16-2Statistics Programs = B + MIn Description 16-4Statistics Programs #$ #$ 16-6Statistics Programs 16-8Statistics Programs %@ 37.9 g &@ XS%@ 40.5 g &@ value 104.5 g %@ 38.6 g &@ 16-10Statistics Programs // / %@ Normal and Inverse-Normal Distributions Logarithmic Exponential Power 16-12Statistics Programs Function ª º ª -+. H % ! 16-14Statistics Programs Variable used temporarily to pass the value S ⋅ 2 π to Inverse programXS@ 16-16Statistics Programs 2f − Grouped Standard Deviation 16-18Statistics Programs Updates ƒxi 2fi in register Increments or decrements N 16-20Statistics Programs / Group%@ @ XU/ %@ 13 g @ / %@ 16-22Statistics Programs%@ %@ Miscellaneous Programs and Equations Time Value of MoneyMiscellaneous Programs and Equations QLI ›LF z …LN `›LI 17-2Miscellaneous Programs and Equations Solve instructions Keys Display Description RPN mode šas17-4Miscellaneous Programs and Equations Retains 5750 in B calculates monthly interest rate 17-6Miscellaneous Programs and Equations Prime Number Generator LBL Y #$ 17-8Miscellaneous Programs and Equations No test is made to ensure that the input is greater than 17-10Miscellaneous Programs and Equations Appendixes and Reference Page Support, Batteries, and Service A-1 Support, Batteries ServiceCalculator Support Answers to Common Questions 2Support, Batteries, and Service Environmental LimitsChanging the Batteries To install batteries Support, Batteries, and Service A-3 4Support, Batteries, and Service Testing Calculator OperationDo not mutilate, puncture, or dispose of batteries Fire. The batteries can burst or explode Support, Batteries, and Service A-5 Self-Test™Ÿ˜—š 6Support, Batteries, and Service Warranty Service Support, Batteries, and Service A-7Europe Country Telephone numbers 8Support, Batteries, and Service Asia Pacific Country Telephone numbersAmerica Country Telephone numbers Support, Batteries, and Service A-9 Regulatory InformationCanada Japan 10Support, Batteries, and Service User Memory and the Stack Managing Calculator MemoryUser Memory and the Stack 2User Memory and the Stack Resetting the Calculator Default Clearing MemoryCategory Disabling Operations Status of Stack LiftNeutral Operations 4User Memory and the Stack EQN FDISP Vlabel nnnn Status of the Last X Register 6User Memory and the Stack€ θ ,r ALG Summary About ALGALG Summary C-1 Doing Two-number Arithmetic in ALG Power FunctionsSimple Arithmetic 2ALG Summary Percentage Calculations ‘27 Q…ALG Summary C-3 Quotient and Remainder Of Division Permutations and Combinations16.12 T 4ALG Summary ALG Summary C-5 Keys Display Description 6ALG Summary Reviewing the Stack ALG Summary C-7 Coordinate Conversions Integrating an Equation Q3 z8ALG Summary ALG Summary C-9 Operations with Complex NumbersTo view the result of complex operations Complex Operations 2 ›3 ` ` Q2 › 10ALG Summary ºy4 ÃË2 Ë5 ¹cº  ºy3 ÃË ALG Summary C-11 12ALG Summary Initial x, y Corrected x, y ALG Summary C-13 20 Page More about Solving How Solve Finds a RootMore about Solving D-1 Function Whose Roots Can Be Found 2More about Solving Interpreting Results More about Solving D-3Example An Equation With One Root Example An Equation with Two Roots 4More about Solving / More about Solving D-5/ -%. `’1.5 LX 6More about Solving Example Discontinuous Function More about Solving D-7 − 1= 8More about Solving When Solve Cannot Find a Root Example a Relative Minimum More about Solving D-9 Example An Asymptote 10More about Solving #LX q LX ›3 ``… ⎟ x + 0.3 − 0.5 =More about Solving D-11 Example Find the root of the equation 12More about Solving !1%ª1%-22Example a Local Flat Region More about Solving D-13 Round-Off Error 14More about Solving Underflow More about Integration How the Integral Is EvaluatedMore about Integration 2More about Integration Conditions That Could Cause Incorrect Results ≥0∞ xe −x dx LX ` LX z‡ 4More about Integration More about Integration 6More about Integration Conditions That Prolong Calculation Time . 8More about Integration More about Integration Page Messages F-1 Messages Or program. This might take a while 2Messages # variable Messages F-3 Self-Test Messages 4Messages Operation Index Name Keys and DescriptionOperation Index G-1 j ˜or —h 2Operation Index Polar to rectangular coordinates Operation Index G-3 4Operation Index P Operation Index G-5 M GComplex Change sign 6Operation Index›Complex …Complex Subtraction Operation Index G-7 RHyperbolic 8Operation IndexVRadians to degrees DSE variable Mvariable Natural exponential Operation Index G-9Wlabel ?Fractional part GTO label Vlabel 10Operation Indexlabel Nnnn Operation Index G-11 ”I”Input variable ‰variable Integer part ISG variable Lvariable LBL label •label 12Operation IndexNatural logarithm Common logarithm Permutations of n items Operation Index G-13FPause WDegrees to radians 14Operation Index Operation Index G-15 16Operation Index Operation Index G-17 18Operation Index ≠0? 13-7 Operation Index G-19If x=0, skips the next program Line ≤0? 13-7 11-10 Given an x-value 20Operation Index Power Returns y raised to the xth power Index-1 Index Index-2 Index-3 Index-4 Index-5 Index-6 Index-7 Index-8 Index-9 Index-10 Index-11 RPN Index-12 Index-13 View Index-14 Index-15

Case Where No Root Is Found

Example: A Relative Minimum.

Keys:

Description:

6 zLX 

More about Solving D–9

6 zLX